Proposition for a Goldbach’s conjecture demonstration Denise Vella-Chemla June 30, 2018

One tries to demonstrate Goldbach’s conjecture. One defines 4 variables : Xa (n) = #{p + q = n such that p and q odd, 3 6 p 6 n/2, p and q primes} Xb (n) = #{p + q = n such that p and q odd, 3 6 p 6 n/2, p compound and q prime} Xc (n) = #{p + q = n such that p and q odd, 3 6 p 6 n/2, p prime and q compound} Xd (n) = #{p + q = n such that p and q odd, 3 6 p 6 n/2, p and q compound} In the following, one notes E(x) the integer part of x (i.e. bxc) and π(x) the number of prime numbers lesser than or equal to x. We have the equality above : it follows from recurrence demonstrations that can be found in [DV] a note written in octobre 2014 ∗ . Xd (n) − Xa (n) = E(n/4) − π(n) + δ(n)

(1)

δ(n) takes values 0,1 or 2. Simplification of note [DV] propositions provided by Alain Connes in may 2018 : [(1) results from the very general fact on any subsets and intersection and union cardinalities : #(P ∪ Q) + #(P ∩ Q) = #(P ) + #(Q)

(2)

Here neglecting limit cases that contribute to δ(n)), one sees that (a) #(P ∩ Q) corresponds to Xa (n). (b) #(P ∪ Q) corresponds to E(n/4) − Xd (n). (c) #(P ) + #(Q) corresponds to π(n). Then we have a very simple proof of (1) as a consequence of (2).] Let us see now a property concerning Xa (n). We decide to represent compound numbers by gray color and prime numbers by white color. We represent odd numbers between 3 and n/2 by rectangles in the bottom of the drawing above and odd numbers between n/2 and n, complementary to n of numbers from the bottom of the drawing by rectangles in the top of the drawing. Rectangles represent contiguous columns associated to decompositions as two odds’sum, and containing x in their bottom part and n−x their complementary in their top part. Columns are contiguously positioned according to the nature of decompositions they contain (according to their type a, b, c or d). We use those colors : - green for #(P ∩ Q) ; - red for #(P ∪ Q) ; - blue for #(P ) + #(Q) = π(n). ∗. Note [DV] : http://denise.vella.chemla.free.fr/numbers-and-letters.pdf, also on Hal https://hal.archives-ouvertes.fr/hal01109052.

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Xd (n)

#(P ∩ Q) = Xa (n)

P

P Q

Figure 1 : n’decompositions contiguously positionned according to their nature Let us remind the set identity (2) : #(P ) + #(Q) − #(P ∪ Q) = #(P ∩ Q). and let’s replace cardinals by associated variables, we obtain π(n) − E(n/4) + Xd (n) − δ(n) = Xa (n) and we wish to have the insurance that Xa (n) is always strictly positive since it counts Goldbach’s n’s decompositions (as sum of two primes). Although, if we demonstrated that Xa (n) = Xd (n) − E(n/4) + π(n) − δ(n) is a relation always verified, this relation doesn’t guarantee that above a certain integer range, Xa (n) is always strictly positive. We note Credit(n) =

P

(BooleanP rime(x) ∧ ¬BooleanP rime(n − x) ∧ BooleanP rime(n + 2 − x))

36x6n/2

Debit(n) =

P

(BooleanP rime(x) ∧ BooleanP rime(n − x) ∧ ¬BooleanP rime(n + 2 − x))

36x6n/2

We find the following recurrence relation for Xa (n), very accounting : Xa (n + 2) = Xa (n) + Credit(n) − Debit(n) + BooleanP rime( n+2 2 ) Adding the boolean BooleanP rime( n+2 2 ) ensure Xa (n)’s positivity for all 2p with p prime, 2p verifying trivially Goldbach’s conjecture. Except those trivial cases of Goldbach’s conjecture verification, we wish to demonstrate that Xa (n) is always greater than Debit(n). We know that Xa (n) is always strictly positive below 4.1018 (by computer calculations from Oliveira e Silva in 2014). First we explain what ensure Xa (n) positivity for numbers n = 6k + 2. Variables values arrays in annex show that for nearly all n = 6k + 2 (notably in the second array), we have Xa (n) = Debit(n) + (n). (n) has either value 1 (when 3 is a Goldbach’s decomponent of n, 1 being compound, 3 + (n − 3) decomposition is not counted by Debit(n)) or value 0. We see studying Credit(n) and Debit(n) definitions that among prime numbers lesser than n/2, ones are counted by Credit(n) while the others are counted by Debit(n), because all prime numbers lesser than n can’t be simultaneously Goldbach’s decomponents of n. This argument ensure the strict positivity of Credit(n). Let us see now why, in the case in which n is of the form 6k + 2, Debit(n) = Xa (n) − (n) : in such a case, prime numbers of the form 6k 0 − 1 can’t be Goldbach’s decomponent of n because if it were the case, n − x = (6k + 2) − (6k 0 − 1) = 6(k − k 0 ) + 3 would be divisible by 3. Prime numbers x that can be Goldbach’s decomponents of n are thus of the form 6k 0 + 1 ; this fact has as consequance that 2

n + 2 − x = (6k + 4) − (6k 0 + 1) = 6(k − k 0 ) + 3 is divisible by 3 and is thus countable as a debit. We have Xa (n) = Debit(n) + (n), that could implies Xa (n)’s vanishing but the Credit(n) addition, Credit(n) being strictly positive permits to avoid such a vanishing. In the case where n is of the form 6k or 6k +4, one sees that Xa (n) is always strictly greater than Debit(n), what guarantees its strict positivity when one substracts Debit(n) to it. Let us try to explain why this is the case : by its definition, Debit(n) is the cardinality of a subset of the set of cardinal Xa (n) (indeed, Debit(n) counts Goldbach’s decompositions of n = x + (n − x) such that n + 2 − x is not prime) ; if Xa (n) were equal to Debit(n), we would have, from the definition of Debit(n), for all Goldbach’s decomposition of n, at the same time n − x prime and n + 2 − x prime, implying that x + (n + 2 − x) would be a Goldbach’s decomposition of n + 2 (i.e. that all Goldbach’s decompositions p1 + p2 of n would be inherited † as Goldbach’s decompositions p1 + (p2 + 2) by n + 2). But we know by congruences study √ that x is a Goldbach’s decomponent of n if and only if x 6≡ n (mod p) for every p lesser than n. All those incongruences couldn’t be verified all at the same time, on one side by x and n, and on the other side by x and n + 2. This has as consequence that for even numbers n of the forms 6k and 6k + 4, Debit(n) < Xa (n) and it implies, by inheritance from n to n + 2, that Xa (n) is strictly positive for all n > 6.

†. see for instance a october 2007 note, Changer l’ordre sur les entiers pour comprendre le partage des décomposants de Goldbach that can be downloaded at http://denisevellachemla.eu.

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Annex 1a : variables values array for even numbers between 6 and 100 n Xa (n) Credit 6 1 0 8 1 0 10 2 0 12 1 1 14 2 1 16 2 1 18 2 1 20 2 1 22 3 1 24 3 1 26 3 2 28 2 2 30 3 1 32 2 2 34 4 2 36 4 0 38 2 3 40 3 3 42 4 2 44 3 2 46 4 3 48 5 2 50 4 3 52 3 3 54 5 2 56 3 3 58 4 4 60 6 1 62 3 4 64 5 4 66 6 1 68 2 5 70 5 4 72 6 2 74 5 4 76 5 5 78 7 1 80 4 4 82 5 5 84 8 3 86 5 4 88 4 7 90 9 2 92 4 4 94 5 6 96 7 2 98 3 6 100 6 6

Debit BooleanP rime( n+2 2 ) 0 0 1 1 1 1 1 1 1 1 1 1 2 1 3 1 2 1 1 2 3 1 2 2 3 2 1 2 3 4 1 4 3 1 2 5 1 2 3 5 2 3 4 1 4 3 4 4 1 2 7 1 5 2 7 4 1 4 6 3 4

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Annex 1b : variables values array for even numbers between 99 900 and 100 000 n Xa (n) Credit 99900 1655 475 99902 694 731 99904 731 1053 99906 1207 506 99908 622 824 99910 824 1097 99912 1288 484 99914 597 617 99916 617 1435 99918 1600 541 99920 789 601 99922 601 1212 99924 1349 510 99926 636 586 99928 586 1424 99930 1590 538 99932 745 630 99934 630 1107 99936 1229 467 99938 587 859 99940 859 1015 99942 1199 541 99944 676 835 99946 835 1010 99948 1180 630 99950 810 613 99952 613 1089 99954 1194 494 99956 605 660 99958 660 1802 99960 2063 374 99962 618 606 99964 606 1113 99966 1222 565 99968 708 900 99970 900 1009 99972 1190 587 99974 736 601 99976 601 1140 99978 1264 620 99980 801 607 99982 608 1092 99984 1216 475 99986 603 736 99988 736 1596 99990 1855 425 99992 638 650 99994 651 1163 99996 1303 478 99998 605 810 100000 810 1213

Debit BooleanP rime( n+2 2 ) 1436 694 577 1091 622 633 1176 1 597 452 1352 789 464 1223 636 420 1383 745 508 1109 587 675 1064 676 665 1000 810 508 1083 605 399 1819 618 497 1079 708 719 1041 736 477 1083 801 1 484 1089 1 603 477 1642 637 511 1177 1 605 600

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Annex 2 : rewriting rules that permit to find a letter of n + 2’s word knowing two letters from n’s word aa → a ab → b ac → a ad → b

ba → a bb → b bc → a bd → b

ca → c cb → d cc → c cd → d

da → c db → d dc → c dd → d

Annex 3 : parts of words associated to even numbers between 6 and 100 (are only provided the letters concerning decompositions of n as sums of the form p + x with p a prime between 3 and n/2 a a aa ca aca aac caa aca aac a caa a aca ca cac ac cca aa acc ca aac ac a caa ca a cca cc ca acc ac ac cac aa ca aca ca cc aac cc ac a caa ac aa c aca ca ca c cac ac cc a cca aa ac a acc ca ca c cac ac ac c a cca ca aa a a acc cc ca c ca aac ac ac a cc caa ca ca a ac cca cc cc c ca acc ac ac a ac cac aa ca c aa aca ca cc c ca a aac cc ac a ac c caa ac aa c ca a cca ca ca c cc a acc ac cc a ac c a cac aa ac a ca a a aca ca ca c cc c ca cac cc ac c ac c ac cca ac aa a aa a ca acc ca ca c ca c cc cac ac cc a cc c ac a cca ca ac a ac a ca c ccc cc ca c ca a cc c acc ac ac c ac c ac a

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